Bayes Prediction on Optimum SS-PALT in Generalized Inverted Exponential Distribution: A Two-Sample Approach

The generalized Inverted Exponential distribution is considered for the study on Optimum Step Stress Partially Accelerated Life Test (SS-PALT) based on different censoring patterns. The first-failure progressive censoring (FFPC) scheme and their special cases are used in the present study. A two-sample Bayes Prediction Bound Length (TS-BPBL) under SS-PALT on FFPC have been obtained and studied their properties by using different special cases of FFPC. Based on simulated and real data set, the properties of the ML estimates and the approximate confidence length under the normal approximation, also have been studied.


Introduction
The Exponential distribution is one of the widely used models in the life-testing experiment because of its simple mathematical practices and interesting properties. An inverted Exponential distribution was discussed by Lin, Duran, and Lewis (1989) by making a valuable changes in the Exponential distributionand and studied the ML estimation, confidence limits, UMVUE and the reliability function based on complete sample case. The generalization of the Exponential distribution was first discussed by Gupta and Kundu (1999) by introducing a shape parameter and named as the generalized Exponential distribution. Dey (2007) studied the inverted Exponential distribution under the Bayesian viewpoint by using different loss functions.
A little work has been done on the generalized Inverted Exponential distribution. Abouammoh and Alshingiti (2009) introduced the generalized inverted Exponential distribution by appending a shape parameter to the inverted Exponential distribution. They observed that, this distribution originated from the exponentiated Frechet distribution (Nadarajah and Kotz The main aim of the present study, is to get the two-sample Bayes Prediction Bound Length (TS-BPBL) for the generalized Inverted Exponential distribution. The optimum Step-Stress Partially Accelerated Life Test (SS-PALT) situation has been used under different special cases of FFPC. A comparison between TS-BPBL with an approximate confidence length, which is obtained by the method of maximum likelihood estimation with normal approximation have also discussed. The numerical illustration has presented on both the simulated and real data set.

SS-PALT under FFPC
In life testing product experiments, it is much difficult to gather lifetimes of extremely reliable products, having a very long lifespan. Because, under the normal operating conditions, a very few or even no failures may occur within a limited testing time interval. The accelerated life test or the partially accelerated life test criterion is very useful test criterion in such cases and, are provided significant reduction in the time and cost of the experiment.  (2008) and Abdel-Hamid (2009) for the details regarding the ALT and PALT criterion. There are two different methods of stress loading in PALT, named as constant-stress and step-stress. The step-stress PALT is considered in the present study, and it permits the test, to be changed from the normal stress condition to the accelerated stress condition at a pre-assumed time. Srivastava and Mittal (2010) have obtained the optimum step-stress partially accelerated life tests for the truncated logistic distribution under the censored data. Tangi, Guani, Xu, and Xu (2012) presents an optimum design for step-stress accelerated life tests based on Type-I censored data followed the two-parameter Weibull distribution. Recently, Prakash (2017 a) discussed about some bound lengths based on One-Sample Plan for Burr Type-XII distribution under the step-stress partially accelerated life tests. Kamal, Zarrin, and Islam (2013) discussed some properties for the inverted Weibull distribution with application of constant-stress partially accelerated life test with Type-I censored data. Hyun and Lee (2015) presents some discussion on Bayes estimation followed the constant-stress partially accelerated life test for the log-Logistic distribution with censored data.
In SS-PALT, all n (say) test units are first run at the normal stress condition up to stress change time (say ), and if it does not fail, then the test is changed to the accelerated stress condition and retained the test until all the units fail. If β is considered as the acceleration factor, then the tampered random variable model under SS-PALT for the lifetime of a unit (Y say) is defined as In the first-failure life test, researcher splits the test units into a number of groups and each groups are assembly of test units. Run all the units simultaneously by first-failure occurred in each group. Wu and Kus (2009) joined this with the Progressive censoring and named as first-failure progressive censoring (FFPC). The FFP censoring scheme has advantages in term of reducing test time, in which more items are used but only a few items are failed. Let us assume (n×k) live test units, in which k independent groups having n units within each groups are putting on a life test. Suppose that, X R 1 < X R 2 < ... < X R m are the progressively firstfailure censored order statistic of size m, with R = (R 1 , R 2 , ..., R m ) pre assumed progressive censoring scheme. Following ) (2017 (b), the joint probability density function under FFP censoring is defined as For k = 1, the Eq. (6) reflects the joint probability density function under the Progressive Type -II censoring. The FFP censoring scheme is combined here with SS -PALT, therefore, the Eq. (6) is re -written as Where, Using Eq. (8) in Eq. (7), the required joint density function based on FFPC on SS -PALT is given as

Maximum likelihood interval estimation
We derive the maximum likelihood estimates using the expectation maximization algorithm and compute the observed information of the parameters that can be used for constructing asymptotic confidence intervals in the present section. The logarithm of Eq. (9) is obtained as Differentiating Eq. (10) with respect to the parameter θ and the acceleration factor β respectively, we get and The second derivatives corresponding to the parameters θ and β are obtained respectively as Asymptotic co-variances and variances of the parameters under ML estimation are obtained from the inverse of the Fisher information matrix. The exact mathematical expressions are not possible to get, however, a numerical method have applied here for the numerical findings.
The inverse of the Fisher information matrix is defined as The second order derivatives involved the unknown parameters, so the Fisher information matrix can be obtained by replacing its ML estimators. A (1 − τ ) 100% improved approximate confidence limits for the parameters θ and β are obtained by following Meeker and Escobar (1998), and given respectively as and Here, Z τ /2 is the percentile of the standard normal distribution with right -tail probability τ /2. Here, the variances of the maximum likelihood estimatorsθ M L andβ M L are the principal diagonal elements of the matrix I respectively. The ML estimators and approximate confidence limits cannot be solved analytically; a numerical method applied here to solve these equations.

Two-sample Bayes prediction bound length
In this section we consider Bayesian inference to get the Bayes Prediction Bound Length under Two-Sample approach (TS-BPBL). The two-sample approach predict the j th order statistic from future sample, based on an informative sample (See Prakash (2015) for more details).
It may be noted that, if both parameters σ and θ are unknown, no joint conjugate prior exists. In such a situation, there are some ways to choose the piece-wise independent priors. In present discussion, gamma prior have been selected for these parameters and given as and A vague prior is assumed here for the acceleration factor β, so that the prior do not play any significant roles in the analyses. The vague prior for the parameter β and the joint prior distribution both are given respectively as and Thus, the marginal posterior densities corresponding to the parameters θ, σ and β are obtained and given respectively as and where Using the Bayes predictive density h Θ (y|x) ; Θ (= θ, σ, β) , for the future observation Y, the cumulative density function is obtained as If Y j denotes the j th order statistic in future sample m (1 ≤ j ≤ m) , then the probability density function is defined for the j th order statistic as The lower and upper Bayes prediction bound limits corresponding to the parameter Θ (= θ, σ, β) for the j th item are obtained by solving following equalities and where Similarly, if l 2m and l 1m are the upper and lower limits for the largest future observation, then the TS-BPBL for the largest future observation (e.g.,j = m) is obtained from

Numerical analysis on simulated data
A simulation study has been performed in the present section for the analysis of the proposed methods. The Monte Carlo simulation technique was used for generating 10,000 FFP censored samples for each simulation by using algorithms described by Balakrishnan and Sandhu (1995).
The samples were simulated for n = 30, m = 10, 15 with hyper-parametric values α = γ (= 0.25, 0.75, 1.25, 2.00, 5.00) and different values of k given in Table 1. All the special cases of FFPC have considered in this section for the analysis.  The values of the parameters under consideration were assumed here as (θ, σ) = (2.01, 0.39), (3.50, 3.67), (5.02, 6.98). The selection of these values meets the criterion that the variance should be unity. The optimal change stress time is calculated by the method of minimizing the asymptotic variance of ML Estimation of the parameter θ and the acceleration factor β. The asymptotic variances of these parameters are calculated from the diagonal elements of the inverse of Fisher information matrix given in Eq. (13). The optimal change stress time is the value, which minimizes asymptotic variance of ML Estimate and, obtained by using Wolfram Mathematica software 10.0. Table 2, presents the Maximum likelihood estimate of the parameter θ and the acceleration factor β. The assumed values of the acceleration factor for the numerical findings are β(= 0.50(0.10)2.50) and it is observed that the ML estimate, first increases as β increases and reaches maximum when β = 1.40 and then decrease, however the magnitude is nominal for both the parameters. Hence, the results are shown here only for the β = 1.40.
It is further noted that, the maximum magnitude was noted for FFP censoring scheme over the other one, whereas the complete sample case shows second maximum estimate as compared to other censoring schemes. It is also observed that, as the parameter (θ, σ) increases, the ML estimate of the parameter θ was increased, however the ML estimate of σ has not been shown similar tendency. Also, as the censored sample size increases the estimate of the parameters increases. Table 3, presents the approximate confidence lengths for the parameter θ and the acceleration factor β under the normal approximation, given in Eq. (14) and Eq. (15). It is observed that the length increases as the confidence level or the censoring size increases. The maximum lengths, noted in the FFP censoring pattern, whereas the minimum length were observed in the complete sample case. The increasing trend in length was also seen first when parameter β increases up to (β = 1.40) , and then decreases. Table 4 -6 presents the Two-Sample Bayes prediction bound length (TS-BPBL) for the parameters θ, σ and β respectively for the assumed parametric values. The maximum bound lengths have been noted in the prior parametric value α = 1.25 = γ and parametric values (θ, σ) = (3.50, 3.67). Other properties have been seen similar as discussed above. Hence, based on above parametric values, it is observed that the FFP censoring scheme provides wider length as compared to progressing or conventional censoring pattern.
Based on above assumed parametric values, the numerical findings were presented in the Tables 7 -9. The similar increasing trend have noted for the parameter β up to (β = 1.40) . Hence, all the results have presents here only for (β = 1.40) .
All the properties have seen similar as discussed above. However, the magnitude of the ML estimate were smaller whereas the TS-BPBL become wider. This shows that the above approach presents good numerical finding under the real data set.

Conclusion
The generalized Inverted Exponential distribution is taken here for the study on Optimum Step Stress Partially Accelerated Life Test (SS-PALT) based on different special censoring patterns of the first-failure progressive censoring (FFPC). The maximum likelihood estimate, approximate confidence limit under the normal approximation and two-sample Bayes Prediction Bound Length (TS-BPBL) have been obtained and studied their properties by using simulated and real data set. One may preferred the FFP censoring pattern over the progressing or conventional censoring based on above selected parametric values.